Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí Department of Mathematic dành cho các bạn yêu thích môn toán học đề tài: GBRDs with block size three over 2-groups, semi-dihedral groups and nilpotent groups. | GBRDs with block size three over 2-groups semi-dihedral groups and nilpotent groups R. Julian R. Abel Diana Combe School of Mathematics and Statistics The University of New South Wales NSW 2052 Australia diana@ Adrian M. Nelson William D. Palmer School of Mathematics and Statistics The University of Sydney NSW 2006 Australia adriann@ billp@ Mathematics Subject Classifications 05B05 20D15. Submitted Sep 9 2009 Accepted Jan 27 2011 Published Feb 14 2011 Abstract There are well known necessary conditions for the existence of a generalized Bhaskar Rao design over a group G with block size k 3. We prove that they are sufficient for nilpotent groups G of even order and in particular for 2-groups. In addition we prove that they are sufficient for semi-dihedral groups. Key words Generalized Bhaskar Rao design. 2-groups. Nilpotent groups. Semidihedral groups. Normal subgroups. Hall-Paige Conjecture. 1 Introduction Definitions and Notation Throughout this paper G is a finite group written multiplicatively 0 ị G is a zero symbol and v b r k x are positive integers with v 3. We denote the cyclic group of order n by C n . A group is a p-group if the order G pr for some prime p and integer r. A group is elementary abelian if it is the direct product of cyclic groups of order p for some prime p. A group is nilpotent if it is the direct product of Pi where each Pi is a pi-group for some prime pi. The trivial group or subgroup is the group with only one element. THE ELECTRONIC JOURNAL OF COMBINATORICS 18 2011 P32 1 Groups with more than one element are non-trivial. A subgroup is a proper subgroup if it is strictly smaller than the whole group. There are several infinite families of groups of particularly importance in this paper. Each of them has a normal cyclic subgroup of index 2 and this can lead to added complications when using normal subgroup constructions of designs. We recall their definitions here and .
